A turntable that plays all types of music well

Weirdly, my Linn/Akito rig will not play polka...just won't...

Lol!

The two LPs just didn't have the same thing on offer for the cartridge to pick up?

I have a decent collection of 50s-70s jazz records. You can hear the sound quality change as they moved from tubes to transistor, from live to studio mixes, and from both to digital. Comparing the acoustic of a 1958 Doc Evans to a muddled digital cacophony is apparent once both the recording technology and the method of creation change.

I highly recommend the Thales TTT-Slim II (or TTT-Compact II if you’d like to go a little above budget). It is the finest made turntable I have ever come across, more finely manufactured than turntables that other builders list for significantly higher prices. Built by a very intelligent engineer trained as a Swiss watchmaker. Hard to top that, and the Slim comes in under $10k.
Turntables don’t effect genres...I’ll just say the Slim spins the records at the right pace and uses battery power to ensure as little deviation in power supply and as little vibration as possible. There are great reviews online.
My second choice would be a Brinkmann Bardo or Taurus.

In case to be genre universal rather play with cart, arm and phono preamp. But I think the topic is subjective and the question is more about your personal preferences of listening then some kind of general rules for different genres. Tables themselves are little to non relevant with genre type.

I like to use dining tables for sit-down dinners, kitchen tables for lunch and snacks, turntables for music. Each of five different turntables for jazz, blues, rock, classical, and swing. Five arms per turntable allows optimization for sub-genres of each. Detachable head shells are of course a must, as how else can you be sure of using the right cartridge for each LP? There should of course be five phono stages (per turntable), and I find a walk-in closet works great for hanging the 50 or so interconnects needed to go with all of this.  

Then we have the amplifiers, we need five of each type- tube and ss, SET and Push-pull, Class ABCDExpialidocious, and then back to the walk-in for speaker cables. Cable elevator- does not reach the top floor.    

This is all for 33. For 45, double everything. 78? Don't ask!